Who, and what, are the "chosen people"?

The "chosen people" are, as you mention the Israelites.

Deuteronomy 7:6
For you are a people holy to the LORD your God. The LORD your God has chosen you out of all the peoples on the face of the earth to be his people, his treasured possession.

This simply means that God has chosen these people and set them aside to be holy.

History

It all began with Abraham:

Genesis 12:1-3 (NIV)
The LORD had said to Abram, “Go from your country, your people and your father’s household to the land I will show you.

2 “I will make you into a great nation, and I will bless you; I will make your name great, and you will be a blessing. 3 I will bless those who bless you, and whoever curses you I will curse; and all peoples on earth will be blessed through you.”

That was the original "chosing", if you will. After that came Isaac and Jacob, who was called Israel.

With Israel (Jacob), God renewed the blessing above (in Genesis 28), saying specifically (in that passage):

Genesis 28:15a
I am with you and will watch over you wherever you go

A man becomes a nation (more history)

God was faithful to his promise to Abraham and Jacob. He made a great nation out of the Israelites and turned them into a nation while they were enslaved in Egypt.

When he freed them from Egypt, he lead them to the base of a mountain and established a few ground rules (the 10 commandments). Later (not much later), he established an entire set of rules that can be found in Leviticus.

These set of rules were the original covenant. The essence of it can be seen in Leviticus 26, but it basically says:

"If you follow my decrees and obey my commands, I will bless you. If you turn away from me, I will curse you."

Summary (What it means to be chosen)

God set the nation of Israel aside. They would be his people and he would be their God.

If they followed him, they would be blessed. If they turned from him, they would be cursed.

Christians are not the Israelites. Christians are not the "chosen people".

Christians are chosen

Having said that, Christians are, undoubtedly, chosen (although we are not "The Chosen People").

John 14:15-24 (NIV)
“If you love me, keep my commands. 16 And I will ask the Father, and he will give you another advocate to help you and be with you forever— 17 the Spirit of truth. The world cannot accept him, because it neither sees him nor knows him. But you know him, for he lives with you and will be[a] in you. 18 I will not leave you as orphans; I will come to you. 19 Before long, the world will not see me anymore, but you will see me. Because I live, you also will live. 20 On that day you will realize that I am in my Father, and you are in me, and I am in you. 21 Whoever has my commands and keeps them is the one who loves me. The one who loves me will be loved by my Father, and I too will love them and show myself to them.”

22 Then Judas (not Judas Iscariot) said, “But, Lord, why do you intend to show yourself to us and not to the world?”

23 Jesus replied, “Anyone who loves me will obey my teaching. My Father will love them, and we will come to them and make our home with them. 24 Anyone who does not love me will not obey my teaching. These words you hear are not my own; they belong to the Father who sent me.

If you read through that, what you'll see is that Jesus has set us aside and given us the same type of covenant that you see mirrored in Leviticus (and Deuteronomy, for that matter). Specifically:

John 14:15
"If you love me, keep my commands."

He goes on to say that he will be part of us and we will be part of him. Clearly, we are "chosen" in this sense.

Summary #2

Christians are "chosen". This means we live in Jesus and we are a part of Jesus and we will inherit eternal life. He has chosen us for eternal life. However, we are not the "chosen people". (See the first half of this answer for that.)